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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution</dfn> Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_9.html">
\begin{equation}
({\bf A}-r{\bf I}) \vec{\xi}={\bf 0}.\tag{6.2.3}
\end{equation}
</div>
<p class="continuation">We require that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_9.html">
\begin{equation*}
|{\bf A}-r{\bf I}|=0 \to 
\left|
\begin{array}{ccc}
1-r &amp; 1 &amp; 2\\
0&amp; 2 -r &amp; 2\\
-1 &amp; 1 &amp; 3-r
\end{array}
\right|=0 \to (1-r)(2-r)(3-r)=0 \to r_1=1, r_2=2, r_3=3.
\end{equation*}
</div>
<p class="continuation">For <span class="process-math">\(r_1=1\text{,}\)</span> from (<a href="" class="xref" data-knowl="./knowl/eq7_9.html" title="Equation 6.2.3">(6.2.3)</a>),</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_9.html">
\begin{equation*}
\left(
\begin{array}{ccc}
1-r_1 &amp; 1 &amp; 2\\
0 &amp; 2-r_1 &amp; 2\\
-1 &amp; 1 &amp; 3-r_1
\end{array} 
\right)
\left(
\begin{array}{c}
\xi^{(1)}_1\\
\xi^{(1)}_2\\
\xi^{(1)}_3
\end{array}
\right)={\bf 0} \to 
\begin{array}{c}
\xi^{(1)}_2+2\xi^{(1)}_3=0\\
\xi^{(1)}_2+2 \xi^{(1)}_3=0\\
-\xi^{(1)}_1+\xi^{(1)}_2+2\xi^{(1)}_3=0 
\end{array} \to \xi^{(1)}_1=0, \xi^{(1)}_2=-2\xi^{(1)}_3.
\end{equation*}
</div>
<p class="continuation">Choose <span class="process-math">\(\xi^{(1)}_3=1\text{,}\)</span> we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_9.html">
\begin{equation*}
\vec{\xi}^{(1)}=\left(
\begin{array}{c}
0\\
-2\\
1
\end{array}
\right).
\end{equation*}
</div>
<p class="continuation">For <span class="process-math">\(r_2=2\text{,}\)</span> we find</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_9.html">
\begin{equation*}
\vec{\xi}^{(2)}=\left(
\begin{array}{c}
1\\
1\\
0
\end{array}
\right),
\end{equation*}
</div>
<p class="continuation">and for <span class="process-math">\(r_3=3\text{,}\)</span> we find</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_9.html">
\begin{equation*}
\vec{\xi}^{(3)}=\left(
\begin{array}{c}
2\\
2\\
1
\end{array}
\right).
\end{equation*}
</div>
<p class="continuation">Thus, the general solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_9.html">
\begin{equation*}
{\bf x}=C_1 \left(
\begin{array}{c}
0\\
-2\\
1
\end{array}
\right) e^{t}+
C_2 \left(
\begin{array}{c}
1\\
1\\
0
\end{array}
\right) e^{2t}+
C_3 \left(
\begin{array}{c}
2\\
2\\
1
\end{array}
\right) e^{3t}.
\end{equation*}
</div>
<span class="incontext"><a href="sec6_2.html#p-264" class="internal">in-context</a></span>
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